"Golden" mirror-symmetrical digit-to-analog converter (DAC)

"Golden" resistive divisor. In the present page we will show the application of the ternary mirror-symmetrical number system for design of digit-to-analog converters (DAC). It is well known that the resistive divisors with the "binary" ratios are the basis of design of the classical "binary" DAC.

And now we will show how design the "golden" resistive divisor with the "golden" ratios of the kind Such resistive divisor is shown in Fig. 1-a.

Figure 1. The "golden" resistive divisor with the transmission coefficient t².

The "golden" resistive divisor in Fig.1-a consists of the "horizontal" and "vertical" resistors of two values R and tR. The connection points A, B, C, D, E connect the resistors of the divisor. In Fig. 1-a it is shown the 5-digit resistive divisor with the connection points A, B, C, D, E. The connection point C corresponds to the 0-th digit. The equivalent electrical circuits of the divisor are shown in Fig.1-b, c, d. The right end part of the divisor regarding to the connection points D, E has the form in Fig.1-b. Let's calculate the equivalent resistance of the parallel joining of the resistors R and tR in Fig.1-b:

(1)

But the same value has the equivalent resistance of the divisor to the left regarding to the point B. The equivalent resistance of the circuit in Fig.1-b regarding to the point D is equal to

If we take in the resistive divisor in Fig. 1-a some arbitrary "horizontal" resistor R with connection points A (the left connection point) and B (the right connection point) then it is easy to show that according to (1) the equivalent resistance of the left-hand and right-hand parts of the divisor regarding to the connection points A and B respectively is equal to t ^-1R.

Then we can represent the equivalent circuit of the divisor for some arbitrary "horizontal" resistor R as it is shown in Fig. 1-c. The latter allows calculating the voltage transmission coefficient from the connection point A to the connection point B:

(3)

Let's consider some arbitrary point C of the resistive divisor in Fig.1-a. Then by using (2) it is easy to show that equivalent resistance of the divisor chain to the left and to the right about the arbitrary point C is equal to tR. Then the equivalent circuit of the divisor regarding to the point C may be presented in the form in Fig 1-d. By using the equivalent circuits in Fig.1-d it is easy to calculate the equivalent resistance of the circuit in Fig.1-d as the parallel joining of the 3 resistors tR, R, and tR:

(4)

Thus the investigation of the divisor in Fig.1-a permits formulating the following electrical properties of the divisor:
1. the voltage transmission coefficient between some neighboring connection points of the divisor (like the points A and B) is back proportional to the square of the Golden Section t²;
2. (2) the equivalent resistance of the divisor in the arbitrary connection point C is constant and equals to ½R.

Note that the property (1) is of a great surprise! Really, it is impossible to imagine that the simplest electrical circuit in Fig. 1-a contains in itself the square of the golden ratio! But this fact stresses one more the fundamental character of the golden ratio, which appears unexpectedly in the simplest electrical circuit!

"Golden" mirror-symmetrical DAC. The "golden" resistive divisor in Fig.1 can be used for design of the "golden" mirror-symmetrical DAC in Fig.2. The latter consists of the fifth (n in the general case) digits. The middle point C corresponds to the 0-s digit a₀ of the input "golden" mirror-symmetrical code a₂ a₁ a₀, a_-1 a_-2 (a_m a_m-1 ... a₀, a_-1 a_-2 ... a_-m in the general case) of the number N. The ternary digits a_i (i = 0, ±1, ±2, ..., ±m) are controlled by the special circuit I₀ connected to the corresponding connection points of the "golden" mirror-symmetrical divisor. The special circuit I₀ consists of the standard electrical generator I₀ and the 3-position electrical key, which are controlled by the ternary digits a_i according to the following rule. If a_i = 1 then the standard electrical current is switched on to the corresponding point of the "golden" mirror-symmetrical resistive divisor in "positive", +I₀. If a_i = -1 then the standard electrical current is switched on to the corresponding point of the "golden" mirror-symmetrical resistive divisor in "negative", -I₀. At least, if a_i = 0 then the standard electrical current I₀ does not be switched on to the corresponding connection point.

Figure 2. The "golden" mirror-symmetrical DAC.

The "golden" mirror-symmetrical DAC has two mirror-symmetrical outputs, U₁ and U₂. Taking into consideration basic properties (1) - (4) of the "golden" mirror-symmetrical divisor, one may show that the mirror-symmetrical outputs U₁ and U₂ are expressed, in dependence on the input "golden" mirror-symmetrical code a_m a_m-1 ... a₀, a_-1 a_-2 ... a_-m, in the following form:

(5)

Let's consider one more the "golden" mirror-symmetrical DAC in Fig.2. The fundamental check property of the latter is the equality

This equality confirms the correctness of the functioning the "golden" mirror-symmetrical DAC. But the violation of the equality (6) is the error indication in the DAC. Thus the "golden" mirror-symmetrical DAC in Fig.2 is the self-checking DAC permitting to realize continues checking the DAC functioning according to (6).

Thus we made one more the small discovery in the field of DAC. We showed that by usage the ternary mirror-symmetrical representation we can design the highly unusual DAC allowing continuously to check the errors in DAC!